On the sphere \(S^{d-1}\) , \(d\ge 3\) , the solution to Problem REF  is known for \(1\le N\le d\)  (basic result) and \(N=d+1\) , see the works by Su [1]} and the author [2]} showing the optimality of a regular simplex. Also, for \(N=2d\) , the optimality of a regular cross-polytope \(\overline{\omega }_{2d}\)  was shown among centered configurations by Boyvalenkov, Dragnev, Hardin, Saff, and Stoyanova
[3]}. A configuration \(\omega _{2d}\subset S^{d-1}\)  is called centered if there is a point \({\bf y}\in S^{d-1}\)  such that \(-\frac{1}{\sqrt{d}}\le {\bf y}\cdot {\bf x}_i\le \frac{1}{\sqrt{d}}\) , \(i=1,\ldots ,2d\) .
